12 Chapter^1we see that any complex number can be expressed as a linear combination,
with real coefficients, of the complex units u 1 and u2; (1.2-3) also shows
that any imaginary complex number can be expressed as the sum of a real
complex number and a pure imaginary number.
In practice, the real complex unit u 1 = (1, 0) is represented by 1, and
the pure imaginary unit u 2 = (0, 1) is denoted by i (sometimes by j). With
this notation (1.2-3) becomes
(a, b) =a+ bi (1.2-4)
The right-side of (1.2-4) will be called the binomial form of the complex
number.
If z = (a,b) =a+ bi, we shall often use the notation a = Rez and
b = Im z as an abbreviations for the real component and for the coefficient
of the imaginary component of z, respectively.
Multiplying the imaginary unit i = (0, 1) by itself, we obtain
i^2 =(0,1)(0, 1) = (-1, 0) = -1 (1.2-5)
which shows that the square of the imaginary unit is the real number -1.
If now we multiply the complex numbers a+ bi and c +di as in elementary
algebra, and then replace i^2 by -1, we obtain
(a+ bi)(c +di)= (ac - bd) +(ad+ bc)i (1.2-6)
which is the correct value for the product according to Definitions 1.1.
Historically, (1.2-6) was obtained first (rather than the other way around)by assuming the existence of a "fictitious" number i such that i^2 = -1.
1.3 Complex Conjugates
Definitions 1.2 Given the complex numberz = (x,y) = x + iy, the com-
plex conjugate of z, or simply, the conjugate of z, denoted z, is the complex
number
z=(x,-y)=x-iy (1.3-1)
The mapping h : z -t z is called complex conjugation.Theorem 1.1 Complex conjugation has the following properties:
- z 1 = z 2 implies that z 1 = z 2 , and conversely.
2. z = z.
- z = z iff z is real.
- z + z = 2x = 2Rez.
- z - z = 2iy = 2i Imz.
- zz = x^2 + y^2 •